We study Laurent polynomials in any number of variables that are sums of at most $k$ monomials. We first show that the Mahler measure of such a polynomial is at least $h/2^{k-2}$, where $h$ is the height of the polynomial. Next, restricting to such polynomials having integer coefficients, we show that the set of logarithmic Mahler measures of the elements of this restricted set is a closed subset of the nonnegative real line, with $0$ being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure $1$ is determined by its $k$ coefficients.

We prove Lawton's conjecture about the upper bound on the measure of the set on the unit circle on which a complex polynomial with a bounded number of coefficients takes small values. Namely, we prove that Lawton's bound holds for polynomials that are not necessarily monic. We also provide an analogous bound for polynomials in several variables. Finally, we investigate the dependence of the bound on the multiplicity of zeros for polynomials in one variable.

Given relatively prime polynomials f(x) and g(x) in ℤ[x] with non-zero constant terms, we show that for n greater than an explicitly determined bound depending on f(x) and g(x), if the polynomial f(x)xn+ g(x) is non-reciprocal, then its non-cyclotomic part is irreducible except for some explicit cases where a known factorization of f(x)xn+ g(x) can easily be described. Prior work of a similar nature is discussed which shows under similar circumstances the non-reciprocal part off(x)xn+ g(x) is irreducible. The current paper establishes and makes use of a result which shows that a reciprocal polynomial f(x) with a root off the unit circle must have a root bounded away from the unit circle by an explicitly given function of the degree of f(x), the leading coefficient a of f(x) and the discriminant of f(x). Notably in this result, a need not be 1.

Several authors, e.g., [D, Sch08, Sch07a, Sch07b], studied the so called reduced length of a polynomial. For a polynomial P it is defined by l(P ) = inf L(PG), where G runs through all monic polynomials in C[x]. In [Sch08], A. Schinzel stated one of the unresolved questions relating to reduced length as: Does the inequality L(P ) ≥ 2M(P ) hold for every polynomial P ∈ C[x] that has a zero on the unit circle? In the cited paper Schinzel proved this inequality for several particular cases and showed that in the general case L(P ) ≥ √ 2M(P ). The purpose of this paper is to prove that in the general case we have indeed L(P ) ≥ 2M(P ).

We find a lower bound On the absolute value of the discriminant of the minimal polynomial of an integral symmetric matrix and apply this result to find a lower bound on Mahler's measure of related polynomials and to disprove a conjecture of D. Estes and R. Guralnick.

We prove that if f(x) = Sigma(n-1)(k=0)a(k)x(k) is a polynomial with no cyclotomic factors whose coefficients satisfy a(k) equivalent to 1 mod 2 for 0 <= k < n, then Mahler's measure of f satisfieslog m(f) >= log 5/4 (1-1/n).This resolves a problem of D. H. Lehmer [12] for the class of polynomials with odd coefficients. We also prove that if f has odd coefficients, degree n - 1, and at least one noncyclotomic factor, then at least one root a of f satisfiesvertical bar alpha vertical bar > 1+log 3/2n,resolving a conjecture of Schinzel and Zassenhaus [21] for this class of polynomials. More generally, we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials, where each coefficient satisfies a(k) equivalent to 1 mod m for a fixed integer m >= 2. We also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy a(k) equivalent to I mod p for each k, for a fixed prime p. Last, we prove that the smallest Pisot number whose minimal polynomial has odd coefficients is a limit point, from both sides, of Salem [19] numbers whose minimal polynomials have coefficients in {- 1, 1}.

For a certain class of functions f: Z --> C an upper bound is obtained for the sum SIGMA(n = a+1)a+H f(n). This bound is used to give a proof of a classical inequality due to Polya and Vinogradov that does not require the value of the modulus of the Gauss sum and to obtain an estimate of the sum of Legendre symbols SIGMA(x = 1)H ((Rg(x) + S)/p), where g is a primitive root of the odd prime p, 1 less-than-or-equal-to H less-than-or-equal-to p - 1 and RS is not divisible by p.

For a certain class of functions f : Z → C f:Z \to C an upper bound is obtained for the sum ∑ n = a + 1 a + H f ( n ) \sum \nolimits _{n = a + 1}^{a + H} {f\left ( n \right )} . This bound is used to give a proof of a classical inequality due to Pólya and Vinogradov that does not require the value of the modulus of the Gauss sum and to obtain an estimate of the sum of Legendre symbols ∑ x = 1 H ( ( R g x + S ) / p ) \sum \nolimits _{x = 1}^H {( ( {R{g^x} + S} )/p} ) , where g g is a primitive root of the odd prime p , 1 ≤ H ≤ p − 1 p,1 \leq H \leq p - 1 and R S RS is not divisible by p p .

Mahler's measure of a monic polynomial is equal to the product of modules of its roots which lie outside the unit circle. By classical theorem of Kronecker it is strictly greater than 1 for any polynomial that is not a product of cyclotomic factors. In this case a number of lower bounds of the measure, depending either on the degree of the polynomial or on the number of its non-zero coefficients, has been found. Here is given an improvement of the bound of the latter type previously found by the author, A. Schinzel and W. Lawton.

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